Effect of finite spectral width on the modulational instability of Alfvén waves

The effect of finite spectral width on the modulational instability of Alfvén waves described by the derivative nonlinear Schrodinger equation is investigated using a method developed by Alber to derive a transport equation for the spectral density. The dispersion relation for a monochromatic wave is regained for a delta spectrum. It is shown that the growth rate and domain of modulational instability diminish as the spectral width increases for both the Gaussian and uniform spectrums.